THE FREE SET PROPERTY FOR CALIBRATED IDEALS Jind rich Zapletal - PDF document

THE FREE SET PROPERTY FOR CALIBRATED IDEALS Jindˇ rich Zapletal University of Florida Czech Academy of Sciences joint with Marcin Sabok and Vladimir Kanovei 1

A book to appear. Kanovei, Sabok, Zapletal: Canonical Ramsey theory on Polish spaces Cambridge Tracts in Mathematics final version date end September 2012 2

Canonization of equivalence relations Given a Polish space X , a σ -ideal I , and a Borel (or analytic) equivalence relation E , is there a Borel I -positive set B ⊂ X such that E ↾ B has a simple form? 3

Possible outcomes • Best: E ↾ B is either identity or B 2 (total canonization); • total canonization for simple equivalences (e.g. classifiable by countable structures); • canonization up to a known set of obstacles– such as E ↾ B is either identity or B 2 or E 0 ; • canonization down to a Borel complexity class–such as E ↾ B is smooth; • Negative: E ↾ B maintains its complexity on all Borel I -positive sets. 4

The free set property Definition. I has the free set property if for every analytic I -positive B and every analytic set D ⊂ B × B there is a Borel I -positive free set , a set B such that D ∩ B × B is a subset of the diagonal. Example. The meager ideal on 2 ω does not have the free set property. ( D = E 0 ) Example. The σ -ideal generated by compact subsets of ω ω does have the free set property. (Solecki-Spinas) Fact. The free set property imples total can- onization for analytic equivalence relations. 5

Calibrated ideals Definition. A σ -ideal I on a Polish space X is calibrated if for every closed I -positive C and closed I -small D n : n ∈ ω there is a closed I - positive C ′ ⊂ C \ � n D n . Example. The meager ideal is not calibrated– let the sets D n enumerate a countable dense subset of X . Example. The ideal of countable sets is calibrated– the set C \ � n D n is positive and contains a perfect subset. 6

Examples of calibrated ideals Class 1. σ -ideals with covering property –every positive analytic set contains a closed positive subset. The ideal of countable sets, the ideal of sets of σ -finite packing measure mass, the ideal of sets of extended uniqueness; σ -ideals obtained from class 1 by Class 2. taking the subideal σ -generated by closed sets. The σ -ideal generated by closed Lebesgue null sets. Other: the σ -ideal σ -generated by Class 3. closed sets of uniqueness Class 4. The σ -ideals with stratified calibra- tion: the σ -ideal generated by closed subsets of [0 , 1] ω of finite dimension. 7

The main theorem Let I be a σ -ideal on a compact Theorem. metric space X , σ -generated by a coanalytic collection of compact sets. If I is calibrated, then I has the free set property. 8

Corollaries for this class of σ -ideals A. Total canonization for analytic equivalence relations. B. Silver property for Borel equivalence rela- tions E : either there is a Borel I -positive set of pairwise inequivalent elements, or the whole space decomposes into countably many classes and an I -small set. C. If Borel E has an I -positive set consisting of pairwise inequivalent elements, then it has a Borel such set. D. The same for finitely many Borel equiva- lence relations simultaneously. 9

Canonization of other objects Example. ( I =ideal of countable sets.) If G ⊂ 2 ω × 2 ω is a graph then there is a perfect set P ⊂ 2 ω such that G ↾ P is either P × P minus the diagonal, or empty. Example. ( I =the σ -ideal generated by closed null sets.) There is a Borel function f : 2 ω × 2 ω → 2 ω such that for all Borel I -positive sets B, C , f ′′ ( B × C ) = 2 ω . 10